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Topics in String Theory | Lecture 3

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I mentioned last time that the horizon of the black hole is not a place where anything varied armed and dangerous is happening a place anything very traumatic would happen to somebody who fell in fact they don't be black hole now baby lets a really big really be black hole what would you find if you fell through the horizon of the black all the answers you a former find nothing there it would look exactly like flat space almost exactly like flat space at the horizon was big enough then they would look like empty space but well so alive because mathematics that we wrote that looks like it does have something striking going on the horizon and can I wanna resolve that a little bit of mathematics mathematics is about at the edge of what I do in this class is not terribly Holland but we a follow-through and unravel what happens Villarreal what the horizon of the black holes really like what's going on there but so that's right down the metric cover black coal Kirk said that's a beginning always starter in the starting point for is right there in that ideally right D house squared with Palestinians for proper time I will use other convention other conventions justice Cheney the sign of the whole thing and call proper dissidents all explained moment a distance squared is just the negative of the proper time between 2 points I'm just doing that partly from variety partly because keep me sane because I know all of these formulas and was formed remember them all we wrote because school it was just something involving deep-sea squared right bt squared in flat space in fact it was just he squared minus BX square why square was easy square that was squared not less than positive is Dedie is bigger towel forward trajectory the time difference is bigger than the space different for example of the object was standing perfectly still In space NDX would be absolutely 0 we've defined the proper time because squared so that it would be part of positive the square of it would be positive but however we were studying a space like distance distance between 2 points at a fixed time With the original definition this would be negative and we took the square root of it we would say that the distance between the pointers imaginary what really gets stuck 1 way or another if we defined we have been very distance so the square of it is positive for time like things that could have be negative for space like things and vise versa so what people those used to different conventions the whenever you don't like your convention you Jesus I'm a time that I don't like my conventions change the sign and the defying yes square was as stand for spatial distance rather than trying but it's just the negative of the proper time square property that it is a big deal but let's right there on the magic of the short chilled black called missed a patient it's 1 minus G are sorry for my aerial guess squared has a minus Plus which AT the sign from the proper time that's the spatially the opposite a minor year DT squared and then goes here 1 over 1 minus 2 and are br square and then another piece here which I'm going to ignore it this had to do with the angles on the sky was our squared time something maker square but let's ignore them now what's not worry about angular distance on sky was just worry about trying and radio distance plus a bit more when your far from the black hole that means when authorities is large large compare what twangy called it's a short fertility the Schwartz showed radius of the black or Ng if I would put in speeds of light it would be to Angie overseas squared right now equal to 1 Otto and Jesus toward children years and if you are far away from the black hole are being much bigger than the Schweiger radius and this is negligible this is negligible is just peachy squared minus minus squared plus squared exactly as it would be in flat space possible Mark as you cluster by called you come to the point where all our is approximately 20 in other words a distant more or less the same as the Schwab children and there's something rather dramatic happens to these coefficients this won't become 0 and this 1 because a nominee gets be 0 but 0 on the nominating something's infinite sure sounds like something unusual was happening by right the media show you exactly what's happening there isn't a great big nothing nothing is happening there it's an artifact of the Cold War of the choice of quark show you some examples and then worked out in detail what's happening at the horizon Oracle only here I want discussed cordon and for the moment now I'm only talking about ordinary Cartesian coordinates on the plane on the blackboard let's call them Hawaii an ex later on the vertical axis will become the time access but for now the vertical axis is just the Y axis I know what the rhetoric of the x y plane that's pretty easy as just squared is equal to the X squared plus BY squared and nothing takes place especially interesting and the origin or anyplace else now let's go the polar coordinates polar coordinates means we characterize a point by a distance which I won't call are because they don't want to confuse it with the sorrow here of different then let's call it rolled the Greek letter role which is sort of like all are they want see other variables played an angle OK 1st of all what is X in terms of role and freighter just drop us perpendicular acts it is equal to Rome times cosigned favor as the definition of cosine and why is is road sign Everybody agree I know about the metric In terms of raw data supposing we start at this point out here now we make a little excursion over here a little differential displacement will differential displacement corresponds the changing all are a certain amount of what's take the red red line here is Dr Eds they are the other red line here how long is act this distance urges bought about distant a perpendicular to it it
a dictator roh Decatur further all you are the big good again be data corresponds that's data now you're right now with the against Street DES glared His eagled the DR squared euro's glitter is Darelle Bureau br for use we save offer over the bureau bureau squared plus rose squared status quo that's a medical poor accord rips notice has a funny property the fight pretty nothing gets infinite metric but not as funny property that the coefficient of the later becomes 0 0 heroical 0 means moving later you don't get anywhere you don't even a few moments later but why not all because you just not moving much when you move around favor at the origin is a situation also this becomes 0 looks like 1 changes nothing is happening if this 0 was see you will see as we go along with their very very similar kinds of things OK everybody understand that just girl polar coordinates but now let's go to Minkowski space space time of special road instead of using white a vertical axis room use its core capital and not going to coats Malti quote capital and the horizontal axis a roll-call X again but he is replacing know what's the matter now the metric now is is equal come minus DT squared plus DX square visitors two-dimensional where we don't need dimensions Forbis express what's happening here is my decay squared plus Peick squared minus is therefore special relativity can solve that summer wreck and looks sort of like this but only assigned change I'm going to Canal introduced polar coordinates but they called hyperbolic polar coordinates says drawing he'll icon is icons like move on to cope with the origin who want to introduce again in a radio variable or a role which measures distance from the origin but not ordinary distant proper distance From the origin and they kind variable McCall maker here's a relationship is the definition X I gotta public X equals rolled cosine later we replace that by X equals Rahal hyperbolic cosine but now say they will call a homemaker that's a standard terminology for for a hyperbolic angle I would tell you more about hyperbolic angles as we go along if you've forgotten but but just right down the equations of black but this isn't right left about something without well completely parallel toward written up there and see is equal to roll out the hyperbolic of maker right now while hyperbolic sine cosine or whatever they are Commack cosine and sine cosine squared plus sighing squared equals 1 that's a statement of a basically Pythagorean theorem that besides square process squares outside good squared the unit triangle how much this hyperbolic guys what's the equation for them posh glared Omega minors square maybe equals 1 it's form pregnant and money tell you what do work Cusher cinch meaning From the diagram up above we can think of this life is drawing a circle restore unit circle your unit circle draw the angle theta parents X is Durand In terms of course for unit circle was just Exs equal the cosine wiser course we do the same thing except now whether circle but with a hyperbola we take a hyperbola OK resist up at stake hyperbola not X squared plus Y squared equals 1 but x squared mightiest he squared equals 1 of paper hyperbola x squared minus squared equals 1 that looks like this at saga and now take a point unit hyperbola a point and drove the same kind of right triangle but we drove for the year of the circle pickup point drop perpendicular drop perpendicular this side here is hyperbolic scion of the angle and this side here is hyperbolic call sign so what Everything is exactly the same except for a slight change and that's change manifests itself a sign of change in the relation between cosigned and OK now relax Eagles Road costume again equals rose synch Omega almost the same as the equation Thaksin Why they have led some guests were the metric looks like Syria's Diaz square which is just where's right areas via squares minus square plus squared is easy workout is
equal to squared the Omega squared off with a minor sorry plus sign Bureau squares was just compare go up metric squares rose square was rose quit Bureau square was Square D favor Square squared saying things instead of Rose cleared the square where minus sign rose squared the Omega square may like but there is a big difference between Megan favor say there is a periodic variable goes around the circle it goes from 0 to 2 and never gets bigger than of of course jumps back 0 organs from a gun arose from minus infinity weighed down here was up to a year becomes 0 at this point and then goes up the plus infinity so hyperbolic angles unlike ordinary angles goal from innocent people being Corsica see that what happens what happens when all did gets very large looks think about from or what happens when it's very very large what happens the hyperbolic it's very easy to delivers pictured gives big that gives big exponentially fast with Omega so this is very big whereabouts Sage Sanchez another way of looking at robots are and what happens to it as to get big or small not as big yeah I know if this is big and the difference between cost and cinch His only a tiny number 1 the since better yet as big as Koch apart from the difference what sort of classes a 100 billion and synch also has to be a 100 billion your minds will prohibit so when all it's a large cast and become very close to being equal to each other right there is legal way out large hyperbolic angle your way out here way up funds like here where she an Ex almost equal to each other we have here on this hyperboloid so way out of this hyperboloid is well maybe gets a large native those who should be way out on hyperbole end weighed down here it goes combines we want to see the Omega it is a kind of time variable it wasn't clear already you will see that or maybe it is a kind of time variable that keeps kind but keeps time this angular kind of way after normal along these hyperboloid Piper promoters or maybe it is a kind of time Rosa kind of distance and bathymetric squares mine this rose squared the Omega square a we we have worked this out but it the follows a very similar manner the way OK so he something let's keep this in mind I want you to remember this formula from earlier trust remember a blackboard I will come back to short shield combined which were terrific here now this metric with this peculiar a row dependency what space isn't what face time as it has just flat Minkowski space with nothing going on special anyway on it's just we chosen to write it In polar coordinates with the quarter net system has something funny going on the origin but there's nothing funny physically going on here is just a quarter of a gimmick OK what we want to show now is that this electorate closest to the horizon is basically like it takes a couple steps I wish I could do this all in 1 Quick Step but takes a couple of steps but then the hard road resist them on the edge of what I like doing a blackboard OK let's rewrite the magic wand slots multiply right this as all minus 2 overall are just written 1 as Oro Bar bt squared minus Mrs. plus are minus 2 and they are squared is coefficient and this 1 hour encouraged to which I have done nothing simply rewritten a former overture leaders looking now at what I'm interested in is the black hole of varying near the horizon another words are very close to energy Willard those who want to zoom in on Oracle's toe injury and expand out and see what's going on there by making some approximations but approximations which are highly accurate right at our recalls toe injury and basic approximation we're going to make it is In the denominator here would not be moving marches let's suppose here here's articles tortoises are axis to Angie I will not only be moving a little bit just to explore the very very neighborhood of this point so don't make little excursions away from this point but I was not going to change much so I'm going to replace are like to injuries said the same earlier no less a good idea because notice what happens when you move are a little bit easier this thinkin change signed so we better keep it the way there's enough with the toe energy on the nominating that doesn't do anything very interesting as you move oral bit if total energy is a kilometer and you move a centimeter only changes a tiny bit calmer summer this assembly but farmers toward Julie it flip side that's dangerous so let's keep this just the way it
I likewise here I just turned this thing over but it that's a 1st step to make a little bit of approximation at 1st and now the next step is to look at this over here is a look at the bureau squared over here under the door attract I'm going to change variables and the change coordinates from 2 new coordinate with Joe Cole role in such a way that this whole thing here and we'll see how this such a way that this whole thing here is just called Bureau square now than I do about how can I say that all are minors 2 warmer whatever here Osaka our service to be twangy here right grand jury to MGM numerate at 1st How can I do that while I haven't told you relatives yet so intellectually Lorelle is no contradiction just maybe that role was related tho in such a way that makes sense volatility and figure out what the relationship between Roh and are the 1st step is dealing with this term he but says that bureau's squared must be the same which is another way equals equals 2 em GE although are minus where take the square root of both sides square square root square this is an equation that determines role in terms of however find role in terms of our from requires markets you integrate both sides of the equation I'm so what do you get on the left-hand side to integrate the left hand side rural we where we get on the right inside that's a little harder not tumor charter let's take out they gather factor to injury and now becomes DAR although square root of all minus 2 Ng this place is not purchased constant me but it's normal to do and with American minutes from with summary of course if a tow your arrears to firms are swear armored where G but see into brokers shorts strike the correct equation Rome musical to square root of 2 and times another factor of tool that makes this kind square root of all minus toe injury if normally differentiate role with respect of and check misses the same equation as saying that euro by DR is equal to 2 ng of armor and toe injury is the solution differentiate with respect our dairy reproduce that that's an exercise for you answer what role has to a order that we can take this Messier and his right bureaus Square is not nice we now have to worry about any factors multiplying this but it may be getting a little complicated because what they want to rewrite everything in terms of role were changing variables and here are appears Neisen simply write about not so bad but there's store energy trial something is equal to 0 roh squared square this equation of square at all hours richer alleged rose squared His equal 8 M jury times are minus 2 was Welker enamel or minus 20 years is just a square a was so bad and now it like the metric this to it's immaterial forever but the library photo Ng here because of the need for various 16 and squeegee squared 16 and G. square as I said a little bit tedious but not too bad OK so that's you always have here we have a roast minus squared DTC glared provided by 16 M Square in G squared plus that's what take a look at this 1st look at best and look best they're not the same but the fairly close you have bureau squared and you have rose squared times the DP square but you have this nuisance 16 and squeegee square downstairs right that doesn't appear over here but well it's really easy get rid of it I do it now we say that we change agreed with change are variable Kuralt now change the key variable toll Amiga agree Watson about year hall that now I just say but TV or G call grab from they go DPP squared 16 M Square square is the Omega square it's just a change of variables by Reese scaling the time by this factor for MG we get there and we get this is equal minus 0 squared the
Omega squared plus Bureau square another words we get exactly flat space In hyperbolic Court now was the exact because we didn't make a slight approximation but an approximation which was essentially exact at equals to injuries were placed an offer by a 2 injury here and are right to Angie here but that's OK because will only moving armor tiny amount away from Wenger so as saying we seemed about to Ojai approximation of the region near the horizon near Oracle's to and has a metric where is it well it has a magic which is just the magic of flat space is good enough to tell us that there's nothing fancy going on the horizon nothing good dramatic no forces become infinite is nothing traumatic happens somebody who find themselves change awkward and we went along a road on but got no dude Back then a broke for a growth is in ready to roll have brought change but they give give up or is slightly larger for recovery is spotted he rose 0 loss is a get back on their feet it is not that far the reproduce would beer yet saying things not Jap minute that OK let's remind ourselves workers quarters look like a drawn over he Ebony redraw the simply has he space for light cone and offers a hyperbolic Angolan measures time so speak along there Mandela but really would put altogether it's just space in some funny coordinates what's this point of right all the here In terms are that point is of course will equal 0 at point is of course role equal 0 bowlers role 0 relic of 0 is all are equals To Wednesday he said could imagine somebody coming in from far away as far away from the black hole moving in moving in moving in moving in and getting to that point and that according to use the horizon of the black hole bachelor the horizon right at that point now I can stop I understand a little better our it's a little weird we start where the black hole which looks like a Brown was far from the black hole looks just like FoxBASE close to the horizon it also looks like flat space but very very different ways the kind of their becomes a hyperbolic angle times foreign G around on transformation of means point here Is it nothing special is going on at the Verizon over here OK any questions about all right other where the horizon is now somebody was outside the horizon is ever aura which is larger than toe injury what does that mean in terms of Rome the it means as a positive value of Rose squared it means out here member hyperbole is from minds of constant role here's role will warm value His roller-coaster another value is a bigger value of Rome's for outside the black hole is a variety of this pair of wine that's outside the tobacco outside black hole will expose over here and you want the same thing a message which will continue to move up our word will send the message Allred and that message will eventually pass any 1 of these hyperbole that message light ready and light out and that library will cross any 1 of these Brewers so we imagine a person stationed at each 1 of these hyperbola sending station that the value of all are any value of our where Amir standing still which means that on a fixed Our trajectory which means a affixed wrote trajectory somebody in here send the message can they get a message to somebody far out that digging a message to anybody no matter how far out because this message will eventually crossed for her role of 50 . 1 is a given proper distance from the horizon but yup but distance distance all Europe yeah write a check is a given distance from right there a coordinate transformation between are enrolled wearers we had written down somewhere areas what's what's get back again well I gave her lost something firm square root in the number square root of 8 the energy gap OK as lawyer is up Our is beginning to Ng the square root positive the square is positive role is positive and they each value of role this 1 value of aura rich value of Are you the find 1 value of world without worry about negative values square of each value of rope comes with a value of our swords given value of art is on 1 of his hyperboloid 1 of these barring OK so that's a that's these a facts outside the black called somebody's stationed outside the black hole Lincoln send the message that anybody else no matter how far how they are yet Her role born stroke truth good you're outside the horizon communicating with them that ucla at couple that steps Morocco is now up up up up up up at the stadium the that 3rd it is 1 of the few people that comment on stuff thought they would are there follow a letter said this was the approximation that we made which is an accurate approximation more and more accurate Our becomes culture close Storm but Hooker where Ali I 102 over here could you send the message out tremolite moves Armed 45 degrees trajectories White on white car if somebody is over here can they send the message out to the outside the outside meaning anyways out beyond this sort of like Cone not saw the world divides into a place where you can send the
message out beyond slight call and over the boy you could send messages out of people at positive values of hard and those places where you can that's the definitional horizon the effect of this whole like like like line is the horizon if you find if you fall for earlier years referral mergers of 2 friends thousand Bob Bobbin systems staying outside the black called better and stay at a fixed distance from the black hole of areas to use going along on his trajectory 2 hours the sides of the ball past that . 1 hours have this point over here she can no longer send a message for Bob Bob can't see he became a guy like a furry can see how he can I get a message from Canarsie Bob and no problem is our store here hours looks back on her light cones sees Bob says an asymmetry Alaskan sea Bob Bob Kinsey hours passed the horizon of the black hole but as anything happen at that point now nothing special at this point this was just good old flat space and nothing special have now that he did so by toward us but she could see yes he kept through the race so it could be exist Jerome he stay home he says that what fees outside her feet past yet she can't see her own feed while she's outside the holders he staying outside not very long as a matter of fact you here see your feet right at you and think you could only see your feet after time for To get you ride from your feet by that time however time off to a list of see it falls like her nothing all nothing at all she just she's doing what always does this is a special place now show her head after her feet past the horizon should hire head decide Oh no I am not going to pass through arrives dinner ahead will never see him that's if she Is there a move along 1 of these trajectory and have feet moving here we all want that is not a good thing this is not a good thing look at her and her feet a cell as low as you don't do anything violent with see a new form of a black holes with you would have seen if you had fallen for black or what somebody sees of EU will if that somebody is stationery with respect to the art coordinates that's a different story so let's see what let's see what a person stationery From now on 1 of these trajectories Missouri things unusual about 1 of these trajectories but but sure make curved Was mean occurred but their accelerated they accelerating should this be a surprise is that to stay out of the black hole somebody has to accelerate you accelerating them yeah at least relative your accelerating relative to a freely following exactly so to stay out of the black or it's true that Bob passed to accelerate CBS have a rocket I would if I have before he supported me I would have had jet Mike to keep from falling back right focus nothing nothing terribly unusual by the further away you get the last week's the lesson and the acceleration stability but it's true that bomb has to accelerate the outside Hey let's go about us you let go out and hours falls into a black hole she he's got a jet back she go a hand and she falls in months earlier she doesn't accelerate she was on straight line so there is an asymmetry is a basic physical asymmetry a body has a jet pack Alice Doesn't and House falls behind and horizon over here she can continue to see Bob nothing special happens Bob loses track of her conopsea so as to Apopka it is Bob Wood but could see is light coming through like comes to him on all these like car so is only year he sees Alice over here I was waiting for millennia ago the further he sees Alice Over here goes further PC's Alice over here what happens as part 40 as he looks back from whose position increasing Omega is is getting larger and larger and larger and is looking back because he ever see Alice Falls the black hole horizon not ultimately asymptotically when Bob works the arbitrary future he simply sees Alice getting closer and closer and closer to the horizon but never crossed so from bombs respectively Alice does not across the horizon she just gets closer and closer and closer to the horizon so of getting can can't do it if you like it or not just knows that gets a lot of our eyes in the back of our head those 2 pages for nose is back overhead following her in and they both be looks factory sees them both extremely close down horizon sees asymptotically they get closer and closer but never quite falling through that verizon here but allow us but she said the service she finally obstruction at this point now no structural she just sales through happily happens work so there's already just at this level even nothing from black holes just thinking about accelerated observers who stay outside of the trajectory of outsiders like cars here is really a bit of attention observer who moves are trajectory like this summer hyperbola our has a set of observation that include among them Alice Falls tore the horizon but never sees her go through the horizon 1 indeed what we have to require Canucks the plot link has each born we cannot answer questions were there were quite mechanics alluring down a little bit of quantum mechanics about black holes they're very elementary we derived entropy and those of some of the things about a black hole by saying we up with photons to remember what we did was an age in the formula sold River started to think about of black holes before them if they rose photo Gaylord all along the rights that for let's throw different here straw 1st Oracle's begun little or the larger Eagles won his articles 1 Ecker B 1 kilometer everywhere along with struck sorry this is big will are 1 kilometer from the horizon authorities bigger than 20 G 2 and G plus a comic Fillinger put support his G plus half a kilometer it is still G plus a quarter of comment is to energy plus
an 8th of a common right on here all year although it's in here somewhere but we don't want the there the modeled on the road a case where a lot for a something singular happened at that point but not the main thing is it's not a physical phenomena exploits achieved quart funny and glitch in the quarter to Oconto it will talk about what's behind you and how you think about quarter but for the time being we have everything we wanted bird flu Thailand salaries it's a lost almost What is the proper turned his road times Omega under the fake fur where sir from legal His rose square the Omega square right so that means rolled along 1 of these trajectories fixed wrote a fixed Row row times deal may get as proper time along so it almost the upper triangle with the Rose scale about it's the same thing is talking about circles 5 is angular proper distance along the circle will not quite into its proper distance except they have multiplied by Mayor Berry radius commit proper OK all right that's that's the properties of a black hole authorized and the bottom line is the properties of black coal horizon of boring the warrant somebody will force the rule but they're interesting to somebody who stays outside somebody who stays outside sees their friend do some weird thing isn't down season flat against the horizon and sold as a kind of tension there but some formal paradox just tension in the description of things not disappoint depressed well does Alice really fall through a dozen she really for what's the answer we the question appointed you from America Bob wanted you know she does there is a lot PC file bathhouse opening and reform future but correct that's what that's what Alice would say Bob said I see forever and ever since he just gets lower and lower bowlers bowled over here is not just a he it is a common type up is Koch's together Protem exists to miss out on it but the last future does have a satellite out on what I see is Alice approaching the horizon asymptotically in there as a sort of you different cover story us really fell through blah blah blah youthful those you are we want series Sheila not just Everything slows down everything slows down here and there but it's not profitable questions to ask does this all that really happened it is a proper profitable questions the ask work what observations in what measurements correlation data Bob Bob will do rule as a physicist orders were areas and we could have scored kind of things Alice detects observed so yup that's right ousted by her that overseas water levels at a pace but the this is particularly among be just for a uniformly accelerated observer if we had an observer riding a rocket at hand enough fuel toward who worked to accelerate the indefinitely if moving away then that observe on that rocket would say for all practical purposes it wasn't into the world could not see beyond that and Kasey nothing for through it so you'd say there is a horizon is called an acceleration arrives at the group's not physically different than the black hole remember what Einstein city Williams being gravity 1st understand acceleration to understand how phenomena happened gravitational field 1st understand how they happened In a uniformly accelerated frame of reference well we've learned is that a uniformly accelerated frame of reference and that's what this is the sequence of there is an acceleration horizon so Einstein could have invented the idea of a horizon before there was any notion of a black hole and then if he was smart enough a fee was a smart of Einstein well this probably means that the sum gravitational context which which is authorized show up friends incident we do not believe me bottles he thought they were violated something OK now I want to go turned back before we do a figure break but I work turned back to the discussion that we started last turned about temperature entropy you a bit and the thermodynamic properties of black holes that they have be costly have a temperature and so Colo we'll do that after said minute break at sea maybe I'll get away with black wordlessly may put on my thermodynamic happy now heat what we did last time is reimagined building up the black hole by dropping will drop since December dewdrop drops in the a bathtub and fill up and a travesty having dropped it takes to fill it to a certain level in the black cloak Acer was Bates I'm a folk times each carrying 1 bit of information that we have to put it to create a black hole size what we discovered it is a bid that we drop in increases area of the horizon by 1 plot point was at 1 1 with a grain of salt window do these real calculations which actable quarter wire factor a quarter report probably means that as is that whoever invented the idea of plant Lane was plant simply got it wrong by a factor of war he should have defined before times bigger 1 4 I forget but there on them we wouldn't have this nuisance for hanging around at Dulles to a historical low at a time and plant nothing whatever about black holes Verizon's information black holes and so forth so it is did some dimensional analysis is that he is if it would the define time tow times different in the fall in the bin OK abound 1 of things we did was to calculate the change in energy if we drop in 1 bit again remember would bit means of that means a photon whose Waverly was comparable the home side black hole promises either G change Nora also told U.S. turned that the definition of temperature just go back a step temperature seems like a medieval and naive meaning to say easily understood concept you feel it would be a goes on cable here you measure it With thermometers we all know what meters but the fact extremely sample and somewhat derived concept entropy is an extremely confusing things when people local and thermodynamics for the 1st time in 2 pieces from the middle what the hell is that on his back entropy is very very basic and in some sense simple energy and entropy are really the basis for thermodynamics statistical
mechanics temperatures of the rock concert was an whatever it is a comes and be it has to do with information that has to do with yes-no questions energy let's suppose we all know when energy woes which I suspect most of us have some idea of entropy and energy the definition of temperature is simply energy that liberated are energy cheated the system when you a bad 1 bit of information with the minimum change of energy when you wear 1 that I say minimum may be very nice and argues the computer your erase a bit of information from your computer that bit of information as to go someplace can't disappear because the atmosphere and necessarily a adds a certain amount of energy to the atmosphere of course if you are sloppy in erasing the better you are you do you do with a sledgehammer you can a lot more energy to the atmosphere and that very careful of course will also be aiding more bits of information a very careful to just be race that minimal away then the minimum and energy that you have to head to the atmosphere in that case it is called the temperature energy necessarily could change Over the entropy by 1 unit but seat keep that definition the back of your head of what temperature is energy needed the change energy liberated when you in 1 bit Otto system only Alaska back now and say we have are black hole where the girl a time was Waverly is comparable to the size of a black hole true and that's calculates the change in energy always the novel is how much energy 1 photon of a wavelength a way of life the wavelength is comparable which wants radius of course aura it's equal to twice the mass times Jerry that's also the wavelength a four-time me or my wavelength is a little bit long shot up a little bit something like that but so what energy of photon of wavelength lambda a usual definition of a wavelength usual symbol for for was the energy of it the energy of is plants constant divided by land that and now speed of light the speed of light as the numerator for the moment keep the constants later on today and throw away the constant on normal way work in units in which the recalled 1 of board can Iramir with keep for a time being by H. fantasy about a bio vs. Jecmenek sent by the energy of a photon went Planck's constant is Calista 0 gets smaller and smaller that makes sense a single photon has a tiny bit of energy why there's a tiny bit of energy because plants constant is so small it's also true that Waverly gets longer the energy of a photon goes down long wavelength photons of low-energy short wavering for farms have high energy the speed of light you could just check forgery up B there OK let's energy you when you win 1 photon which carries 1 bit well the energy they had 1 bit at temperature so this must be the temperature of a black hole and when I say temperature I mean within a factor of maybe just to wasn't parties figures were not keeping track of a detailed commerical constant I'll tell you numerical constants later if you want the temperature is H. C. I always in some keeping track for a moment the moment from keeping track of worth something out of here see squared speed white square your lesser temperature but remember that land that is supposed to be chosen equal to the size of the black holes what but better you are all our H To see all over twice and Jean and squared enumerator plants constant times CQ and we I can trust factor to with factor too is not trustworthy up reauthorize answers in a minute 5 CQO divided by the mass times G most want curious then that as the man's gets bigger and bigger although several things 1st thing to peers that means that would temperature of the black Eliza quantum effect it would be easier if age bar of talks constant were 0 so it's a field which only happens we call us Planck's constant is now 0 it's another way of saying it's very because of quantum mechanics C-Cube laugh and therefore of eventual reasons the man's is in the denominator that's interesting the bigger the less of a black hole colder readers that's only on for the following reasons is energy he equals M C squared solar energy mass are the same thing you use of the ideal you energy to something gets harder on black hole score when your energy source bigots may the cooler it better than not consequence will come to that odd consequence in a moment but was just keep track of it this is the temperature Of the black hole or give you the right formula now variety age bars and I know the answer and a pipe that's the actual correct age barred the time C-Cube divided by a pint Times man turned deserve Newton's constant OK 1st question how hot is any real mats Blackall so that we would stick in a nest from my what's the mess of a storm of the 30th kilograms of something the big number downstairs G is a small number C-Cube is huge in each bar is key he teeny our both competition big numbers of small numbers but believe final answer is this is a small number is but not at all well it's more misses about us all of us black hole for black hole of stellar cast is about can have a minus 8 Calvin so called anything is cold anything man people talk about getting down Fernando Calvin's for some reason or other maybe things in laboratories were actually cold but this is damn called its so-called that it's not too much colder than empty space even in the remotest regions of the universe where a far far from many stars anything else we know that the temperature on these bases to the extent that empty space bases-empty ordinary empty space it is about 3 degrees Kelvin that's the microwave background the black hole cold this black hole is colder than anything in empty space that means even if you had such a black hole which was nothing else around the fall into it even if you manage to create a black hole in otherwise completely empty space the empty space around it would be harder than it is what happens when you pick a cold object and put it on a high after bath he flows from the Colorado from me warm region for the call region another way of saying it is the black hole simply swallows up movie of microwave radiation just suck but you could also just think of it as the flow of heat from on the call OK what happens now why happens if I'm outer space called the somebody brings a heaters warned me up the car I get warmer I'd absorbs he think about what happens black all the black hole and absorb some heat from its surroundings what happens to its massive and absorb some heat it increases its energy increases but then what happens with temperature it goes that gets even call it gets even Coahuila gets even called it even becomes more
efficient and absorbing heat but absorb heat gets called Catch-22 organs of the moral absorbs heat to call the reject but let's imagine that we had a black hole where it came from which happens to be a little bit warmer In the year surroundings which way it is the ego he goes from a black hole into the atmosphere the black hole loses mast what happens when I was mess it gets harder so run away situation for black owes cooler than its environment it will absorb energy called it the black holes warmer than its environment it give off energy get hotter odors run as it as a Jew running away of course in getting hotter it gets mess is getting smaller and smaller threw an explosive situation mascots more smaller it gets harder and harder and eventually get so hot that just explore it that if he runs in that direction so a black called In thermal equilibrium with its environs is unstable even if it had exactly the same temperature as its environment a flocculation could happen by accident just by random fluctuations in might absorb a little bit of extra energy if it did so we get cold I call that it would be more an absorber I will run away all of my accidentally give off a little extra energy which case we get hotter and they give off yet more so nets and by calls I'm spending might think that means there and physical there are other systems in nature which have this property in fact any system which held together by gravity star stars when they won the four-star gives up energy 0 we shrinks as collapses when it gives off energy try it thinks cool I'm cool because it did because thanks the cold the thinks it has less pressure the stars don't think but need a less pressure all the stuff out because of the pressure stuff fast forward and what happens when the falls In roughly speaking you could just imagine that the gas and stuff and I started getting
squeezed what happens when you squeeze yet so starter an example of a system held together by gravity which our when it when it gives up energy Gitana Stalin thermal equilibrium with its environment thermal equilibrium its environment would mean that the environment half have the temperature of the stock so we are talking about the real universe but prehistoric we tried to put the thermal equilibrium of its environment would do the same thing we're also run away no new specific right exactly Vegas Pacific he is unstable but that doesn't mean by coast exist in means in particular rise in the real world there simply be absorbers which was slowly absorb radiation and eventually the universe will expand the cool and the temperature will go down below the temperature of a black hole in which case the black hole stopping off energy and will evaporate OK so let's let's talk about evaporation while young may have temperature because they have temperature their black bodies black bodies not the sense that they are completely dark but the sense of give loft thermal radiation I will go through the there are just a very simple derivation of a rate at which black holes evaporate rate which they lose their energy but do it he units since I really don't want to try the remember the places where C H bargee come into this will work in units wet Everything is water and they will convert back the sensible units now I always like to say if I were on a desert island that I never physics Booker Nunavut computer anything else what units where work and it would be plant units because those are the only units really remember things up its color filters have some rough idea of how big things are pockets and under the through a delegation of the luminosity of a black hole plant you as I said the only reason it is everything is 1 thing that I have to worry whether constants grow but similar plant units are they units awaits the speed of light plants constant and Newton's constant legal toward Sea whose age Oracle's GE Eagles 1 in those units of ruling plant plaintiff is about 10 to the minus 35 meters the plant tolerated it is about a 10th of the minus 40 43 seconds and the Plonka Mass late kind and mouse that's a complete set of and the plant mass is about 10 minus 84 kilogrammes the plant command is not a particularly unusual mass legacy kilograms is about massive dust mold you could see it with the naked eye so it's a it's a rather ordinary man a plant time incredibly short my point is incredibly small factor clock time is nothing but the kind that it would take for a light gray to cross the plant distance so that really independent if you take the plant distance as Khalil would like to go across it it's plant time us work in those units that now a black hole as temperature because temperatures review anything that has temperature take particular efforts couldn't or vacuum will radiate black body radiation blackbody radiation simply thermal radiation electromagnetic radiation incidentally will also radiate gravitational radiation but the theories are absolutely Parra on let's talk about formula for the luminosity of warm body the luminosity is is the rate at which had energy so we can write it as minors only because the object is losing energy we can ride it as money this year rate of change of the energy of the object some object it is the rate of change of its energy is losing energy so Debye IDT is negative and minus debate defuse possible was that 1st of all it's proportional to area of the object when object radiates a redhead from surface the bigger the area Fassero already area of the object appears to be a area and this will become the area of the horizon of the black hole black holes radiate from eyes and aimed it depends on the temperature obviously the heart of the temperature a faster rate may North formula put the burden the the temperature Quito the 4 famous formula stuff on multiple forms temperature for the full of power you could arrive that from dimensional considerations just dimensional formula and what about the coefficient which appears to be an order of magnitude is 1 because all of the parameters where else could be right plant units and clunky amidst this we rate at which the black hole or anything else would lose energy now energy is Mets Eagles MC Square is 1 so them he is an Millicom right this as the rate of change of the loss of mass greater change of a mass of the year loss of mass per unit time is the area the temperature before Was area be area is proportional to the watch over radius square foot patina some for high but now we don't care firms things what 5 depend are Schwartz chilled squared away about the temperature of doing a stint of loss a formula for the temperature Gorecki notes to define temperature but
I find see 1 a lot of letters bulletin course Ng Art right now I'm we just remotely remind you were temperature lead we found 8 all over land at times see the Swazi energy we 1 bit agency on 1 and served temperatures as while Milan by the land was supposed to be the radius of a black hole temperatures as 1 of older ladies fish watcher lady and appended to the bought that's 1 over radius bought and this whole thing has become a radius 4 or the rate of change of mass with time is equal to 1 OR square all but oryzae in his of young was ours to Geagea's is 1 was so this is also just 1 moment and square remember where every CRU Yucaipa him because proportional to each other and his energy This is a prime rate of change in energy and the armed area where a lot of forces temperature of 4 so we gave us with formula OK let's multiplied by M squared but minors M square near was squad D and Square and my beauty and Square the indicate get apart from a factor of 3 and squared times the NDP is just equal was on the right hand side of the errors equation what happened to right it's 1 right 1 writer other later right there surprise from a factor of 3 years the prime rate of change of the man's queue of the black is equal toward how long does it take a black hole to lose all of its Mets it has a uniform rate at which it loses may you may ask you move is diminishing 1 Nunavut per unit time how long does it take for all that mess start with the disappear and Q and you know right this is a change in masculine delighted by the change in times 1 and therefore the total time it takes the lose all of is just and so a cube that's how long it takes evaporation time we're not sure how long is it promised plug-in what's plug-ins summer sun numbers China so amassed like all our armed forces or and the 30th kilograms so how applicant plant units we did this plant units the plant units so the soil it tainted 30 of kilograms commonly talk masses that think the 38th because plant messes tend to minus 8 kg so take the 30 Qualcomm is a mass of the sort of their services are some will for OK I was the evaporation time well it's and killed service pentameter what's 30 times 314 yards and killed is tender the 114 but tender 114 what plant times want times times deeper locked up plunked I missed and the minus 43 seconds so seconds as Aziz correspond to take over that and the 71 tale of a 71 2 seconds From leaders as 1 years 3 times and was 2nd driver account of the 7th Have a 60 and the 64 years what's that allow us to present data theaters but it and the 10th years right so this is kind the 54 universe ages black holes evaporate slowly the reason of course is a very very cold and you get Tito the so even after the universe cool stood pouring with a black hole will evaporate it's still evaporating very slowly With could I let's take a smaller black hole is but her daughter brought back you but if I had accused Enrico Fermi recover but that you rule us as well as to what the buttocks Don Cheadle on this it did 6 years at 7 7 for used a gun was said but they did histories says air force before and said but you have to
multiply 10th 3 because cubic meters about a thousand their kilograms not young now so tend to attend yes and and and the what's it kilograms and and kilograms it is Our harmony Arab monks tenderly 18th tenderly 18 plant now 3 times 18 per cent of the 54 tender 54 want times and now we figure of 43 There was apparently 11 seconds and the 11 seconds a lot shorter what's tend to be 11 seconds the 4 years and not a not terribly long has stood 3 B by called on a black hole on if you were a kg world few kilograms alive time is a shortage of workers nite and were more workers OK that's luminosity of a black hole if the site of the job but we really small them to that EU radius is proportional to amass right or could be made us Of Thee son is 10 of the 30th and it corresponds to 1 0 0 1 2 kilometers so 1 kilogram would be 10 to the minus 30th kilometers that's pretty small is not Smalls a was pretty small now is given the right he our wants once of course not all of these there are summaries some of these will be North include yacht young but that's where only if their ambient only the ambient temperatures cooler than they are right right so that's an interesting question of what man's With the black hole have temperature which is 3 degrees of background radiation all you have to your work that out what's the size of a black hole size meeting radius and mass the black coal richer Her return 0 3 degrees OK I think we've done both before which means that we have the mechanism furs entropy real weeds a certain sense or not so much guessing most of this is based on very definite calculation but it's the kind of it's the kind of theoretical work where you put some principles together and if you believe all the principles then this it is what has to be conclusion of that there are some small numerous degrees of freedom that are not apparent in the standard theory of black holes on standard theory of gravity the understanding of the now is that the standard theory of gravity is in a certain sense like fluid dynamic fluid dynamics describes a smooth flow of a uniform fluid a also described by our more turbulent flows and so forth but does not take into account be comic structure of words there are of course circumstances where fluids might be atomic structure might become important but ordinary fluid flow the equations of it are effective equations which fell take into account the microscopic structure you don't put entropy into flow but you don't ask where that entropy came from but just the fact is In fluid tells you visit him microscopic structure to close its early on that he that the something there besides just smooth flow something on the line at smaller distances there he numerous degrees of freedom sold as things stand today well as things stood at some point we understand a little more now but we could say the theory there was talk about up till now is a strong team that does more than theory of gravity is a microstructure tour is more water fury of black holes just Einstein's theory something smaller more numerous that we're not keeping track of him No Ordinary General and in that sense it to him to hint of something yet to come we know more about now but let's wait let's wait we'll talk about what the hidden structures might be use of energy is still will be more growth rate it was that bad policy but your grep erect and our rap With the figure that we draw here says things fall through the horizon however doubled get back out if commitment Miller a taste of that is most fundamental amendments so fundamental people and forget about it without it new roles all of the rules of physics would make no sense for example the rules say that in variances of various kinds kind translation invariance leads to energy conservation it does only if you have the rule are due example example right now is a erased slugger along here let things get the microscopic keep everything else that suppose we lived in a world where EU races
really can't arrest because they satisfied reasons emotion that made them come to rest rather than need 30 physics of our armed friction and so for the real fundamental laws emotion and maybe raises come rest well 1st thing is there would not be information conservation white because every comes the rest no matter how it started so with forget would start comes arrest and so would losers the information but we started about energy conservation there's is no energy to be conservators comes the rest from a momentum conservation no momentum conservation Mom so what what is what is drawings with the laws of physics which just heavy races come you violate some of the basic principles of classical mechanics and member of quantum mechanics but and they know what they're violating his information conservation so ideas of energy conservation momentum conservation all conservation all the Salem friendlier combat this comes to rest violates momentum conservation does is Violet translation invariant tabletop imagine the tabletop for you look Foreman went on forever and ever and the laws of physics with other racers Comparex doesn't violate the translation variants so at least statement that momentum conservation follows From now translation variances apparently not true you need something else the other thing that you need this information conservation information conservation plus invariant leads the momentum conservation sorrow got only fundamental probably be most single fundamental idea physics the but now was not the same armed Is it more fundamental cause our knowledge probably at the same level 1 Meda principles which were about which you couldn't begin possibility me causality factory you mean the where the curve yup yup that's it's a similar kind of principled but big game deer lectures classical tests can for and columns of that state's they tainted that's a printer young or in classical mechanics that collegial steer abstract and information conservation in classical mechanics is a statement that the phase space volume doesn't change but you can shrink the phase space volume that In quantum mechanics call you carry and unit carries a principle that orthogonal state stay off final time which means the distinctions are always preserved job be ble connected with energy it is of the this is a part of you can be sure it's got real possibility that it's not operate on probabilities war microscope states the works it is that with a net loss of probable From the beauty get that's thermal equilibrium yacht for more I'm trust yet but that that the issue but that they were to probabilities of buyers can be he he he will indicate that about half of them from probably he also a problem that when I was a little bit of fun to be functional yes nor entropy doesn't apart probability distribution probability distributions on our last best if retailers did might give year not and drippy minus the sum of all configurations the probability of configuration times the library configuration This is 1 number and 1 number can't determine he's determined S S is given in terms of the peat bog growth this is taking us taking us off young on OK for more
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Titel Topics in String Theory | Lecture 3
Serientitel Lecture Collection | Topics in String Theory
Teil 3
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15114
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch

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Fachgebiet Physik
Abstract (January 24, 2011) Leonard Susskind uses the most complex math that will be used in the course with the hopes that it will give a better idea of how a black whole works mathematically. In the last of course of this series, Leonard Susskind continues his exploration of string theory that attempts to reconcile quantum mechanics and general relativity. In particular, the course focuses on string theory with regard to important issues in contemporary physics.

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